Cutting down that offending tree in your yard can be a quick job. Getting rid of its roots may be more of a challenge. The good news is that unless the roots are causing problems with your landscape, you can leave them under the soil. However, if the roots surface under walkways or in your lawn, cracking the pavement or interfering with mowing, they have to go. How you remove them depends on the size of the roots.
Cut down the tree, leaving 3 to 4 feet of trunk above the ground.
Dig a circle with a radius of 3 to 4 feet around the base of the tree using a sharp spade; sever the roots as you dig. Dig at least 1 foot deep to reach all surface roots. If a root is too large to cut with a spade, remove the soil around the root and cut it with a saw or ax.
Attach a winch to the tree trunk and anchor it securely against another tree or a heavy-duty vehicle. Make sure the tree or vehicle is capable of sustaining the pressure from winching the trunk and roots free from the soil.
Engage the winch to pull the tree trunk from the soil. (See Resources for proper winching techniques.) Small and medium trees will pull free with the major roots intact.
Fill the area with new topsoil and seed with grass. You will soon forget a tree ever existed in this place.